Overweight Korean Adolescents and Academic Achievement

Abstract

The relationship between academic achievement and being overweight among South Korean high school students was examined. Data used in the regression were from the Korean Education and Employment Panel Survey. The theoretical framework that poor school performance increases the risk of adolescents’ being overweight, which, in turn, causes poor school performance, was supported. With no other direct or indirect association between weight and achievement, an overweight high school student’s poor performance in school was assumed to be a function of the psychosocial well-being variables and self-concern about weight. A simultaneous-equation regression model that endogenized the likelihood an individual is classified as overweight (a binary variable) and the performance of that individual on the College Scholastic Ability Test (CSAT) incorporates the unobserved psychosocial well-being correlated with both school grades and being overweight.

Appendices

Appendix

Probability-Based Survey Design and Stratification Issues

We modified the Nelson–Olsen estimator to accommodate the stratified survey design of the Korean Education and Employment Panel Survey (KEEP). Researchers use sample designs in the sample selection process to reduce the risk of distorting views of a population because of the variability of attributes among individuals in the population. To make statistically valid inferences for a population, the sample design should be incorporated into the data analysis. Because individuals surveyed have specific probabilities of selection and the probabilities of selection are related to the response variables, an analysis that does not take the different selection probabilities into consideration may lead to biased results (Lohr 1999). In a stratified representative survey such as the present case, the inverse of an individual’s selection probability is the weight attributed to that individual in terms of how many other individuals are in the population with similar attributes or personal characteristics. The sum of the individual weights is an estimate of the population of surveyed individuals.

Weighted least squares (as in the case of Eq. 1) or weighted maximum likelihood estimation produce point estimates which reflect the average relationship of covariates with a given response (in this case, academic performance or the probability of being overweight) for the entire population; e.g. ${\hat{\varvec{{\beta }}}}_{1} = \left( {{\mathbf{X^{\prime}WX}}} \right)^{ - 1} {\mathbf{X^{\prime}Wy}},$ with W a conformable matrix with the survey weights along the diagonal, zeros elsewhere. However, computation of the covariance matrices in the case of least squares (e.g., Eq. 1) or probit regression (e.g., Eq. 2) is slightly more complicated, but possible, given a wide range of software with built-in functions designed to estimate covariance matrices that account for the survey design. To our knowledge, there are no software packages capable of producing the simultaneous least squares-probit model Nelson–Olsen covariance estimator for stratified surveys. We suggest a modification of the Nelson–Olsen covariance estimator that accounts for the stratified probability based on the survey design of our data.

First, we outline the Taylor series expansion theory behind the construction of the least squares and probit models adjusted for stratified survey sample designs, which assume no cross-equation correlation. The linear approximation of the least squares covariance matrix we used is attributed to Fuller (1975), and the covariance matrix we applied for the probit model is attributed to Binder (1983), Lehtonen and Pahkinen (1995), Morel (1989), Roberts et al. (1987), and Skinner et al. (1989).

Least Squares Taylor Series Expansion for Stratified Survey Designs

The linear approximation of the least squares covariance matrix follows Fuller (1975). For the stratified survey sample design, observations were represented by an n by (k + 2) matrix (n and k are observations and model parameters, including an intercept, respectively): (W, y, X) = (w _si , y _si , x _si ), where W (w _si ) is a matrix (vector) of sampling weights (e.g., the inverse of the selection probability in the population); y is the response variable; X is a matrix of exogenous explanatory variables; s = 1,…, S identifies the stratum number, with S total stratum; i = 1,2,…, n _s is the number of observations collected from a stratum; and $n = \sum\nolimits_{s = 1}^{S} {n_{s} }$ is the total number of observations in the sample. Additional dimensions are possible for cluster sampling within stratum.

For the linear regression model, consider the residual vector ${\hat{\mathbf{v}}} = {\mathbf{y}} - {{\mathbf{X} {\hat{\varvec{{\beta}}}}}}_{1},$ where the sith element $\hat{v}_{si}.$ We first computed the 1 by k row vectors,

$$ {\hat{\varvec{{\xi }}}}_{1}^{si} = w_{si} \hat{v}_{si} {\mathbf{x}}_{si},$$

(8)

$$ {\bar{\varvec{{\xi }}}}_{1}^{s} = n_{s}^{ - 1} \sum\nolimits_{i = 1}^{{n_{s} }} {\hat{\varvec{{\xi }}}_{1}^{si} },$$

(9)

and then the k by k matrix,

$$ {\hat{\mathbf{G}}}_{1} = \sum\nolimits_{s = 1}^{S} {\frac{{n_{s} }}{{n_{s} - 1}}} \sum\nolimits_{i = 1}^{{n_{s} }} {\left( {\hat{\varvec{{\xi }}}_{1}^{si} - {\bar{\varvec{{\xi }}}}_{1}^{s} } \right)^{\prime } \left( {\hat{\varvec{{\xi }}}_{1}^{si} - {\bar{\varvec{{\xi }}}}_{1}^{s}} \right)}.$$

(10)

Let (X′WX) = A. Then the covariance matrix for linear least squares estimator for stratified survey data is,

$$ {\hat{\mathbf{V}}}_{1} = Var\left( {\hat{\mathbf{{\beta }}}_{1} } \right) = {\mathbf{A}}^{ - 1} {\hat{\mathbf{G}}}_{1} {\mathbf{A}}^{ - 1}.$$

(11)

This estimator is similar to the Huber–Eicker–White heteroskedastic-robust covariance estimator (White 1980), except that the survey design weights are included in ${\hat{\mathbf{G}}}_{1}.$

Probit Regression Taylor Series Expansion for Stratified Survey Designs

The second order approximation of the covariance estimator we applied to the probit model is attributed to Binder (1983); Lehtonen and Pahkinen (1995), Morel (1989); Roberts et al. (1987), and Skinner et al. (1989). Additional notation helps to review the Taylor series expansion of the probit regression covariance matrix adjusted for analysis of probability based stratified surveys. Maximizing the weighted log likelihood function for the probit regression yields a consistent estimate of the main effects on the response variable, ${\hat{\varvec{{\beta }}}}_{2}.$ Let ${\hat{\varvec{{\pi}}}}_{si}$ be the vector of estimated probabilities of being classified as y _2i  = 1; e.g., ${\hat{\varvec{{\pi }}}}_{si} = \Upphi \left( {{\mathbf{x^{\prime}}}_{si} {\hat{\varvec{{\beta }}}}_{2} } \right),\,{\hat{\mathbf{D}}}_{si}$ the matrix of partial derivatives of the probit log likelihood function with respect to and evaluated at ${\hat{\varvec{{\beta }}}}_{2},$ and

$$ {\hat{\varvec{{\xi }}}}_{2}^{si} = w_{si} {\hat{\mathbf{D}}}_{si} \left( {diag\left( {\hat{\varvec{{\pi }}}_{si} } \right) - {\hat{\varvec{{\pi }}}}_{si} {\hat{\varvec{{\pi }}^{\prime}}}_{si} } \right)^{ - 1} \left( {y_{2si} - {\hat{\varvec{{\pi }}}}_{si} } \right), $$

(12)

$$ {\bar{\varvec{{\xi }}}}_{2}^{s} = n_{s}^{ - 1} \sum\nolimits_{i = 1}^{{n_{s} }} {\hat{\varvec{{\xi }}}_{2}^{si} }. $$

(13)

The probit covariance matrix adjusted for the probability-based survey stratification is therefore,

$$ {\hat{\mathbf{V}}}_{2} = Var\left( {\hat{\varvec{{\beta }}}_{2} } \right) = {\hat{\mathbf{B}}}^{ - 1} {\hat{\mathbf{G}}}_{2} {\hat{\mathbf{B}}}^{ - 1}, $$

(14)

with

$$ {\hat{\mathbf{G}}}_{2} = \sum\nolimits_{s = 1}^{S} {\frac{{n_{s} }}{{n_{s} - 1}}} \sum\nolimits_{i = 1}^{{n_{s} }} {\left( {\hat{\varvec{{\xi }}}_{2}^{si} - {\bar{\varvec{{\xi }}}}_{2}^{s} } \right)^{\prime } \left( {\hat{\varvec{{\xi }}}_{2}^{si} - {\bar{\varvec{{\xi }}}}_{2}^{s} } \right),} $$

(15)

and

$$ {\hat{\mathbf{B}}} = \sum\nolimits_{s = 1}^{S} {\sum\nolimits_{i = 1}^{{n_{s} }} {w_{si} {\hat{\mathbf{D}}}_{si} \left( {diag\left( {\hat{\varvec{{\pi }}}_{si} } \right) - {\hat{\varvec{{\pi }}}}_{si} {\hat{\varvec{{\pi }}^{\prime}}}_{si} } \right)^{ - 1} {\hat{\mathbf{{D}}^{\prime}}}_{si}.}}$$

(16)

The Nelson–Olsen Least Squares-Probit Estimator for Stratified Survey Variance Estimation

We extend the simultaneous least squares probit Nelson–Olsen covariance estimator by noting that Var(Π ₁) and Var(Π ₂) assume the disturbance terms in the reduced-form equations are homoskedastic, and that the constant terms $\left( {c = \hat{\sigma }_{1}^{2} - 2\hat{\gamma }_{1} \hat{\sigma }_{12},\,d = \hat{\gamma }_{2}^{2} \hat{\sigma }_{1}^{2} - 2\hat{\gamma }_{2} \hat{\sigma }_{12} } \right)$ contain information about the population variance of each model, and the covariance between both models. By decomposing c and d to correspond with each respondent (instead of averaging over all individuals as with the usual Nelson–Olsen covariance estimator), and then including these elements in computation of ${\hat{\mathbf{G}}}_{1}$ and ${\hat{\mathbf{G}}}_{2},$ we incorporate information about the cross-equation disturbances into the variance estimators of Eqs. 6 and 7 in similar fashion to the Huber–Eicker–White sandwich covariance estimators. The following notation is useful:

$$ {\mathbf{A}}_{{{\mathbf{H}}_{1} }} = {\mathbf{H^{\prime}}}_{1} {\mathbf{AH}}_{1}, $$

(17)

$$ {\hat{\varvec{{\xi }}}}_{12}^{si} = w_{si} \left( {\frac{{I\left( {y_{2si} = 1} \right)}}{{\phi \left( {{\mathbf{x^{\prime}}}_{si} {\hat{\varvec{{\Uppi }}}}_{2} } \right)}}} \right){\mathbf{x}}_{si}, $$

(18)

$${\bar{\varvec{{\xi }}}}_{12}^{s} = n_{s}^{ - 1} \sum\nolimits_{i = 1}^{{n_{s} }} {\hat{\varvec{{\xi }}}_{12}^{si} },\,{\text{and}} $$

(19)

where ϕ is the normal probability density function, and I(∙) an operator equal to one when y ₂ = 1, zero otherwise (Keshk 2003). Summing over all observations in Eq. 18, and then dividing that total by the sum of the sample weights yields the population covariance estimate ($\hat{\sigma }_{12}$ in Eq. 5 between the structural Eqs. 1 and 2. Decomposing $\hat{\sigma }_{12}$ to match individual observations, it is possible to construct the matrix analogues of ${\hat{\mathbf{G}}}_{1}$and ${\hat{\mathbf{G}}}_{2},$ which in turn effectively models the cross-equation disturbances within each stratum;

$$ {\hat{\mathbf{G}}}_{12} = \sum\nolimits_{s = 1}^{S} {\frac{{n_{s} }}{{n_{s} - 1}}} \sum\nolimits_{i = 1}^{{n_{s} }} {\left( {\hat{\varvec{{\xi }}}_{1}^{si} - {\bar{\varvec{{\xi }}}}_{1}^{s} } \right)^{\prime } \left( {\hat{\varvec{{\xi }}}_{12}^{si} - {\bar{\varvec{{\xi }}}}_{12}^{s} } \right)}. $$

(20)

Given ${\hat{\mathbf{G}}}_{1},$ ${\hat{\mathbf{G}}}_{2},$ and ${\hat{\mathbf{G}}}_{12},$ rewrite Eqs. 6 and 7 as;

$$ V\left( {\hat{{\varvec {\alpha }}}_{1}^{Survey} } \right) ={{\bf A}}_{{{{\bf H}}_{1} }}^{ - 1} \left( {{{\bf H^{\prime}}}_{1}{\hat{{\varvec {\Uppsi }}}}_{1} {{\bf H}}_{1} } \right){{\bf A}}_{{{{\bf H}}_{1} }}^{ - 1} + \hat{\gamma }_{1}^{2} {{\bf A}}_{{{{\bf H}}_{1} }}^{ - 1} {{\bf H^{\prime}}}_{1} {{\bf A}}\hat{\bf V}_{2} {{\bf AH}}_{1} {{\bf A}}_{{{{\bf H}}_{1} }}^{ -1},$$

(21)

$$ V\left( {\hat{\varvec{{\alpha }}}_{2}^{Survey} } \right) = \left( {{\mathbf{H^{\prime}}}_{2} {\hat{\mathbf{V}}}_{2}^{ - 1} {\mathbf{H}}_{2} } \right)^{ - 1} + \left( {{\mathbf{H^{\prime}}}_{2} {\hat{\mathbf{V}}}_{2}^{ - 1} {\mathbf{H}}_{2} } \right)^{ - 1} {\mathbf{H^{\prime}}}_{2} {\hat{\mathbf{V}}}_{2}^{ - 1} {\mathbf{A}}^{ - 1} {\hat{\varvec{{\Uppsi }}}}_{2} {\mathbf{A}}^{ - 1} {\hat{\mathbf{V}}}_{2}^{ - 1} {\mathbf{H}}_{2} \left( {{\mathbf{H^{\prime}}}_{2} {\hat{\mathbf{V}}}_{2}^{ - 1} {\mathbf{H}}_{2} } \right)^{ - 1} $$

(22)

with

$$ {\hat{\varvec{{\Uppsi }}}}_{1} ={\hat {{\mathbf G}}}_{1} - 2\hat{\gamma }_{1} {\hat{\mathbf{G}}}_{12}, $$

(23)

$$ {\hat{\varvec{{\Uppsi }}}}_{2} = \hat{\gamma }_{2}^{2} {\hat{\mathbf{G}}}_{1} - 2\hat{\gamma }_{2} {\hat{\mathbf{G}}}_{12}. $$

(24)

The simultaneous least squares-probit model covariance matrices, adjusted for survey design stratification are, therefore: ${\hat{\mathbf{V}}}_{1} = Var\left( {\hat{\varvec{{\Uppi }}}_{1} } \right)$ (for the least squares equation),${\hat{\mathbf{V}}}_{2} = Var\left( {\hat{\varvec{{\Uppi }}}_{2} } \right)$ (for the probit equation), and $\left( {\hat{\varvec{{\xi }}}_{1},{\hat{\varvec{{\xi }}}}_{2} } \right)$are residuals from the reduced-form Eqs. 3 and 4. In the most general case, $V\left( {\hat{\varvec{{\alpha }}}_{1}^{Survey} } \right)$ and $V\left( {\hat{\varvec{{\alpha }}}_{2}^{Survey} } \right)$ are robust to arbitrary forms of heteroskedasticity, and are applicable beyond the realm of variance estimation of stratified survey studies.